NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA3252 Linear and Network Optimization
Tutorial 1
1. A small bank is allocating a maximum of $30,000 for personal and car loans during
the next month. The bank charges an annual interest rate of 1
Tutorial 3: Outline of Solutions
Q1. (a) For each of the following sets, decide whether it is representable as a polyhedron?
(i) The set of all (x, y ) R2 satisfying the constraints
x cos + y sin 1 [0, /2]
x 0
y 0.
(ii) The set of all x R satisfying the c
Tutorial 5: Outline of Solutions
Q1. Consider a minimization linear programming problem in standard form. Let x be a basic
feasible solution associated with the basis B. Prove the following:
(a) If the reduced cost of every nonbasic variable is positive,
Tutorial 4: Outline of Solutions
Q1. Consider the standard form polyhedron P = cfw_x Rn |Ax = b, x 0. Suppose the
matrix A has dimensions m n and its rows are linearly independent. For each of the following
statements, state whether it is true or false. I
Tutorial 11: Outline of Solutions
Q1. In the following two maximum ow problems with source s = 1 and sink t = 5, determine
all s t cuts. Find the minimum s t cut in each case. Solution:
30
2
2
4
(4,0)
25
20
18
15
1
(4,6)
(0,2)
(2,0)
3
(8,0)
3
1
12
12
(6,0
Tutorial 10: Outline of Solutions
Q1. Adam and Eve are planning a drive from location A to E. The time to travel and the scenic
rating for the roads in the network are given below. All roads are one-way. Adam wants to
reach location E as fast as possible
Tutorial 9: Outline of Solutions
Q1. Consider the following linear programming problem.
min
s.t.
5x1 5x2 13x3
x1 + x2 + 3x3 20
12x1 + 4x2 + 10x3 90
x1 , x2 , x3 0
Let x4 and x5 denote the slack variables for the respective constraints. The optimal tableau
Tutorial 7: Outline of Solutions
Q1. Consider the following linear programming problem:
(P ) min 3x1
+ 4x3
s.t. 2x1 + x2 x3 2
x1 + 3x2 5x3
7
x1 0, x2 0
(a) Write the standard form of the above LP problem.
(b) Verify that the dual of the above LP problem
Chapter 10:
Facts & Tools
#1
a) Private cost
b) External benefit
c) External cost
d) Private benefit
e) Private cost
f) Private benefit
g) External cost
h) External benefit
#2
Yes it might reduce the undersupply of people who get flu shots. As people star
Technological progress: A burden in disguise?
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A preview of emerging technological trends and their societal implications
The idea of putting our planet on life support is a sobering tho
Transmittance
Measurement
Presented by
Dr. Richard Young
VP of Marketing & Science
Optronic Laboratories, Inc.
Optronic Laboratories, Inc.
Outline of Presentation
Types of transmittance:
Regular
Diffuse
Factors affecting measurements:
Regular Transmi
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Enter the following password: MOH
MTHSC 1060 Final Review Study Guide
*Note that these quick notes do not include everything that you need to know, just some of the most important
concepts. *
Chapter 1
-Composite functions from tables. Ex:
x
-1
0
1
f(x)
3
1
0
g(x)
-1
0
2
h(x)
0
-1
0
Use t
8
O
15.999
8 is the _ and signifies the number of _ in oxygen
15.999 is both the _ and the _ of oxygen
_ mass is the weighted average mass of all isotopes of the
element
_ mass is the grams it takes of any element to equal one mole of
that element.
One mo
Decision Analysis
and Tradeoff Studies
Terry Bahill
Systems and Industrial Engineering
University of Arizona
terry@sie.arizona.edu
, 2000-10, Bahill
This file is located in
http:/www.sie.arizona.edu/sysengr/slides/
Acknowledgement
This research was suppor
0113111311 Jllbhlllu, Wu1ua sauna 1.1.1.: nE-_-.
The liable describes a conversation hetwwn Prajapati, the creator god, and
his three species of children: gods, human beings, and demons cfw_also divine crea
tures, though adversaries of the gods). The Deus
Tutorial 2: Outline of Solutions
Q1. (a) Reformulate the following problem as a linear programming problem:
max min(x1 , x2 )
s.t. |2x1 + x2 | 7
3x1 x2
0.5
1 + x1 + x2
x1 , x2 0.
Solution:
max z
s.t. z x1
z x2
2x1 + x2 7
2x1 x2 7
2.5x1 1.5x2 0.5
x1 , x2
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA3252 Linear and Network Optimization
Tutorial 11
1. In the following two maximum ow problems with source s = 1 and sink t = 5,
determine all s t cuts. Find the minimum s t cut in each case.
30
2
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 1: Introduction to Linear Programming
1 / 51
Linear programming is a mathematical modeling technique
designed to minimize or maximize a linear cost function subject to
a nite set of linear equality and inequali
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 2: Geometry of Linear Programming
1 / 30
A polyhedron or polyhedral set can be described in the form
cfw_x Rn | Ax b, where A is an m n matrix and b is a vector
in Rm . Geometrically, a polyhedron is a nite int
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 3: The Simplex Method
1 / 81
Consider the standard form of a LP
min c x
s.t. Ax = b
x 0.
Assume A is an m n matrix and rank (A) = m with m n. Let
P = cfw_x Rn | Ax = b, x 0. If P = , then P has an extreme
point
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 4: Duality Theory
1 / 44
Starting with a linear programming problem, called the primal LP,
we introduce another linear programming problem, called the dual
problem. Duality theory deals with the relation betwee
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 5: Sensitivity Analysis
1 / 27
Sensitivity (or post optimality) analysis deals with the study of
possible changes in the optimal solution as a result of making
changes in the original problem.
Why study sensiti
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 6: Introduction to Network Optimization
1 / 49
Network ow problems are a special case of linear programs and
are among the most frequently solved optimization problems.
Everywhere around us networks are apparen
MA3252 LINEAR AND NETWORK
OPTIMIZATION
Topic 7: The Network Simplex Method
1 / 19
In this topic, we develop the details of the simplex method applied
to network ow problems (known as the network simplex method).
This can lead to a much faster algorithm th
Topic 1
Introduction to Linear
Programming
Example (Matrix Transpose)
Row Column
2 3 4
A=
5 8 9
x1
x2
B=
xn
Column Vector
2 5
A ' = 3 8
4 9
B ' = ( x1
x2 xn )
Row Vector
Example (Column Vectors)
Note :
For this course, an n dimensional vector x i
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA3252 Linear and Network Optimization
Tutorial 2
1. (a) Reformulate the following problem as a linear programming problem:
max min(x1 , x2 )
s.t. |2x1 + x2 | 7
3x1 x2
0.5
1 + x1 + x2
x1 , x2 0.
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA3252 Linear and Network Optimization
Tutorial 3
1. (a) For each of the following sets, decide whether it is representable as a polyhedron?
(i) The set of all (x, y ) R2 satisfying the constraint